Lot scheduling on a single machine to minimize the (weighted) number of tardy orders

https://doi.org/10.1016/j.ipl.2020.106009Get rights and content

Highlights

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    Studying lot scheduling problems with order split and no-split.

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    Proposing a polynomial solution for minimizing the number of tardy orders (split).

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    Providing a pseudo-polynomial solution for the weighted version (split).

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    Proving strongly NP-hardness for the minimum number of tardy orders (no-split).

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    Introducing a heuristic for the no-split problem.

Abstract

We consider a single machine lot scheduling problem. A number of customer orders of different sizes may be processed in the same lot. We consider first the setting that splitting orders between consecutive lots is allowed. We focus on minimizing the number of tardy orders. A polynomial time solution algorithm is introduced for this problem. We then study the extension to minimizing the weighted number of tardy orders. This problem is NP-hard, and a pseudo-polynomial dynamic programming is provided and tested. We also study the setting of no-split. The problem of minimizing the number of tardy orders in this context is proved to be strongly NP-hard, and an efficient heuristic is introduced.

Introduction

In the class of lot scheduling problems, the scheduler/producer receives customers' orders of different sizes which are processed in lots. The total size of the orders processed in a single lot cannot exceed its capacity. Two recently published papers, Hou et al. [2] and Zhang et al. [6], studied lot scheduling problems with the following very realistic features: (i) all the lots have identical size, (ii) all the lots have identical processing time, (iii) the size of a single order cannot exceed the (common) size of a lot, and (iv) an order can be split between two consecutive lots.

Among the many areas of applications of lot scheduling (see [2], [6]), we mention just a few: integrated circuit tests in a semiconductor factory, glue production with a heated container, stone wash processing of textile products, and digital advertisement display in the e-commerce industry. Many scheduling measures appear to be relevant to this setting. Hou et al. [2] solved the problem of minimizing total completion time, and Zhang et al. [6] focused on minimizing weighted completion time. Both considered the case of order splitting. In this paper we study two other (related) measures: (i) minimum number of tardy orders, and (ii) minimum weighted number of tardy orders. For the first, we introduce a polynomial time solution algorithm, which is a modified version of the classical Moore's Algorithm [4]. The second problem is NP-hard, and we propose an efficient pseudo-polynomial dynamic programming algorithm. We also discuss the special case of a common due-date for all the orders. We further extend the problem of minimizing the number of tardy orders to a setting where order split is not allowed. This version is an extension of the bin-packing problem, which is known to be strongly NP-hard [1], and we therefore focus on the introduction of an efficient heuristic.

The paper is organized as follows: Section 2 contains the notation and formulation. Next we focus on the setting that order splitting is allowed: in Sections 3 and 4 we introduce the solution methods for the problem of minimizing the number of tardy orders and for the weighted version, respectively. Section 5 contains the heuristic for the case of no-split. In the last conclusion section, we also suggest topics for future research.

Section snippets

Notation and formulation

We study single machine lot scheduling problems. There are n orders, which are to be processed in lots. The capacity of each lot is denoted by L, and the size of order j is denoted by pj>0;j=1,…,n. We assume that pj≤L,j=1,…,n. The total size of the orders processed in a single lot cannot exceed its size. For the first two problems, we assume that each order can be split if the remaining capacity in the lot is less than the size of the order, and the remaining part of the order may be processed

Minimizing the number of tardy orders (Problem P1)

As mentioned, due to lot availability, for any allocation of orders to lots, the actual completion time of an order is the completion time of the lot to which it was assigned. Thus, the completion time of an order can be one of the following values: iP,i=1,2,…,NL. This leads to a trivial observation, which is relevant to the objective functions considered here: for any allocation of orders to lots, all the due-date values which are between iP and (i+1)P (i.e., such that iP≤dj<(i+1)P) can be

Minimizing the weighted number of tardy orders (Problem P2)

In this section we study the more general setting of lot scheduling to minimize the weighted number of tardy orders. Consider its special case with L=1 (i.e., unit-size lots). This special case is the classical problem of minimizing the weighted number of tardy orders (1||∑wjUj), which is known to NP-hard in the ordinary sense [3]. It follows that Problem P2 is NP-hard as well.

We introduce in the following a pseudo-polynomial dynamic programming algorithm (denoted DP1), thus proving that P2 is

Minimizing the number of tardy orders with no split (Problem P3)

In this section we study the case that an order must be processed entirely in a single lot, i.e., order splitting is not allowed. As mentioned, this problem is an extension of the well-known bin-packing, which is known to be strongly NP-hard; see Garey and Johnson [1]. We therefore introduce in the following an efficient heuristic.

We start by computing the modified due-dates, sorting the orders according to EMDD, and then sorting the orders sharing the same modified due-date according to LOS.

Conclusion

We solved single machine lot scheduling problems to minimize the number of tardy orders. Following recently published papers, we considered first the setting that orders can be split between consecutive lots. The first studied problem was shown to have a polynomial time solution which is an adaptation of the classical Moore's Algorithm [4]. We then considered the weighted version, which is known to be hard even for unit lot size. An efficient pseudo-polynomial dynamic programming algorithm was

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This research was supported by the Israel Science Foundation (grant No. 2505/19). The second author was also supported by the Charles I. Rosen Chair of Management, and by The Recanati Fund of The School of Business Administration of The Hebrew University.

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